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Simple machine

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From Wikipedia, the free encyclopedia
Mechanical device that changes the direction or magnitude of a force
This article is about the concept in physics. For independent record label, seeSimple Machines. For the Internet forum software, seeSimple Machines Forum. For broader coverage of this topic, seeMechanism (engineering).

The six classical simple machines
History of technology

Asimple machine is amechanical device that changes thedirection ormagnitude of aforce.[1] In general, they can be defined as the simplestmechanisms that usemechanical advantage (also called leverage) to multiply force.[2] Usually the term refers to the six classical simple machines that were defined byRenaissance scientists:[3][4][5]

A simple machine uses a single applied force to dowork against a single load force. Ignoringfriction losses, the work done on the load is equal to the work done by the applied force. The machine can increase the amount of the output force, at the cost of a proportional decrease in the distance moved by the load. The ratio of the output to the applied force is called themechanical advantage.

Simple machines can be regarded as the elementary "building blocks" of which all more complicatedmachines (sometimes called "compound machines"[6][7]) are composed.[2][8] For example, wheels, levers, and pulleys are all used in the mechanism of abicycle.[9][10] The mechanical advantage of a compound machine is just the product of the mechanical advantages of the simple machines of which it is composed.

Although they continue to be of great importance in mechanics and applied science, modern mechanics has moved beyond the view of the simple machines as the ultimate building blocks of which allmachines are composed, which arose in the Renaissance as aneoclassical amplification ofancient Greek texts. The great variety and sophistication of modern machine linkages, which arose during theIndustrial Revolution, is inadequately described by these six simple categories. Various post-Renaissance authors have compiled expanded lists of "simple machines", often using terms likebasic machines,[9]compound machines,[6] ormachine elements to distinguish them from the classical simple machines above. By the late 1800s,Franz Reuleaux[11] had identified hundreds of machine elements, calling themsimple machines.[12] Modern machine theory analyzes machines askinematic chains composed of elementary linkages calledkinematic pairs.

History

Engraving from an 1824 mechanics magazine illustrating Archimedes's statement that given a place to stand, with a lever a person could move the Earth

The idea of a simple machine originated with the Greek philosopherArchimedes around the 3rd century BC, who studied theArchimedean simple machines: lever, pulley, andscrew.[2][13] He discovered the principle ofmechanical advantage in the lever.[14] Archimedes' famous remark with regard to the lever: "Give me a place to stand on, and I will move the Earth," (Greek:δῶς μοι πᾶ στῶ καὶ τὰν γᾶν κινάσω)[15][16][17] expresses his realization that there was no limit to the amount of force amplification that could be achieved by using mechanical advantage. Later Greek philosophers defined the classic five simple machines (excluding theinclined plane) and were able to calculate their (ideal) mechanical advantage.[7] For example,Heron of Alexandria (c. 10–75 AD) in his workMechanics lists five mechanisms that can "set a load in motion":lever,windlass,pulley,wedge, andscrew,[13] and describes their fabrication and uses.[18] However the Greeks' understanding was limited to thestatics of simple machines (the balance of forces), and did not includedynamics, the tradeoff between force and distance, or the concept ofwork.

During theRenaissance the dynamics of themechanical powers, as the simple machines were called, began to be studied from the standpoint of how far they could lift a load, in addition to the force they could apply, leading eventually to the new concept of mechanical work. In 1586 Flemish engineerSimon Stevin derived the mechanical advantage of the inclined plane, and it was included with the other simple machines. The complete dynamic theory of simple machines was worked out by Italian scientistGalileo Galilei in 1600 inLe Meccaniche (On Mechanics), in which he showed the underlying mathematical similarity of the machines as force amplifiers.[19][20] He was the first to explain that simple machines do not createenergy, only transform it.[19]

The classic rules of slidingfriction in machines were discovered byLeonardo da Vinci (1452–1519), but were unpublished and merely documented in his notebooks, and were based on pre-Newtonian science such as believing friction was anethereal fluid. They were rediscovered byGuillaume Amontons (1699) and were further developed byCharles-Augustin de Coulomb (1785).[21]

Ideal simple machine

If a simple machine does not dissipate energy through friction, wear or deformation, then energy is conserved and it is called an ideal simple machine. In this case, the power into the machine equals the power out, and the mechanical advantage can be calculated from its geometric dimensions.

Although each machine works differently mechanically, the way they function is similar mathematically.[22] In each machine, a forceFin{\displaystyle F_{\text{in}}} is applied to the device at one point, and it doeswork moving a loadFout{\displaystyle F_{\text{out}}} at another point.[23] Although some machines only change the direction of the force, such as a stationary pulley, most machines multiply the magnitude of the force by a factor, themechanical advantage

MA=FoutFin{\displaystyle \mathrm {MA} ={F_{\text{out}} \over F_{\text{in}}}}

that can be calculated from the machine's geometry and friction.

Simple machines do not contain a source ofenergy,[24] so they cannot do morework than they receive from the input force.[23] A simple machine with nofriction orelasticity is called anideal machine.[25][26][27] Due toconservation of energy, in an ideal simple machine, the power output (rate of energy output) at any timePout{\displaystyle P_{\text{out}}} is equal to the power inputPin{\displaystyle P_{\text{in}}}

Pout=Pin{\displaystyle P_{\text{out}}=P_{\text{in}}\!}

The power output equals the velocity of the loadvout{\displaystyle v_{\text{out}}\,} multiplied by the load forcePout=Foutvout{\displaystyle P_{\text{out}}=F_{\text{out}}v_{\text{out}}\,}. Similarly the power input from the applied force is equal to the velocity of the input pointvin{\displaystyle v_{\text{in}}\,} multiplied by the applied forcePin=Finvin{\displaystyle P_{\text{in}}=F_{\text{in}}v_{\text{in}}\!}. Therefore,

Foutvout=Finvin{\displaystyle F_{\text{out}}v_{\text{out}}=F_{\text{in}}v_{\text{in}}\,}

So the mechanical advantage of an ideal machineMAideal{\displaystyle \mathrm {MA} _{\text{ideal}}\,} is equal to thevelocity ratio, the ratio of input velocity to output velocity

MAideal=FoutFin=vinvout{\displaystyle \mathrm {MA} _{\text{ideal}}={F_{\text{out}} \over F_{\text{in}}}={v_{\text{in}} \over v_{\text{out}}}\,}

Thevelocity ratio is also equal to the ratio of the distances covered in any given period of time[28][29][30]

voutvin=doutdin{\displaystyle {v_{\text{out}} \over v_{\text{in}}}={d_{\text{out}} \over d_{\text{in}}}}

Therefore, the mechanical advantage of an ideal machine is also equal to thedistance ratio, the ratio of input distance moved to output distance moved

MAideal=FoutFin=dindout{\displaystyle \mathrm {MA} _{\text{ideal}}={F_{\text{out}} \over F_{\text{in}}}={d_{\text{in}} \over d_{\text{out}}}\,}

This can be calculated from the geometry of the machine. For example, the mechanical advantage and distance ratio of thelever is equal to the ratio of itslever arms.

The mechanical advantage can be greater or less than one:

In thescrew, which uses rotational motion, the input force should be replaced by thetorque, and the velocity by theangular velocity the shaft is turned.

Friction and efficiency

All real machines have friction, which causes some of the input power to be dissipated as heat. IfPfric{\displaystyle P_{\text{fric}}\,} is the power lost to friction, from conservation of energy

Pin=Pout+Pfric{\displaystyle P_{\text{in}}=P_{\text{out}}+P_{\text{fric}}}

Themechanical efficiencyη{\displaystyle \eta } of a machine (where0<η <1{\displaystyle 0<\eta \ <1}) is defined as the ratio of power out to the power in, and is a measure of the frictional energy losses

η≡PoutPinPout=ηPin{\displaystyle {\begin{aligned}\eta &\equiv {P_{\text{out}} \over P_{\text{in}}}\\P_{\text{out}}&=\eta P_{\text{in}}\end{aligned}}}

As above, the power is equal to the product of force and velocity, so

Foutvout=ηFinvin{\displaystyle F_{\text{out}}v_{\text{out}}=\eta F_{\text{in}}v_{\text{in}}}

Therefore,

MA=FoutFin=ηvinvout{\displaystyle \mathrm {MA} ={F_{\text{out}} \over F_{\text{in}}}=\eta {v_{\text{in}} \over v_{\text{out}}}}

So in non-ideal machines, the mechanical advantage is always less than the velocity ratio by the product with the efficiencyη{\displaystyle \eta }. So a machine that includes friction will not be able to move as large a load as a corresponding ideal machine using the same input force.

Compound machines

Acompound machine is amachine formed from a set of simple machines connected in series with the output force of one providing the input force to the next. For example, abench vise consists of a lever (the vise's handle) in series with a screw, and a simplegear train consists of a number ofgears (wheels and axles) connected in series.

The mechanical advantage of a compound machine is the ratio of the output force exerted by the last machine in the series divided by the input force applied to the first machine, that is

MAcompound=FoutNFin1{\displaystyle \mathrm {MA} _{\text{compound}}={F_{{\text{out}}N} \over F_{\text{in1}}}}

Because the output force of each machine is the input of the next,Fout1=Fin2,Fout2=Fin3,…FoutK=FinK+1{\displaystyle F_{\text{out1}}=F_{\text{in2}},\;F_{\text{out2}}=F_{\text{in3}},\,\ldots \;F_{{\text{out}}K}=F_{{\text{in}}K+1}}, this mechanical advantage is also given by

MAcompound=Fout1Fin1Fout2Fin2Fout3Fin3…FoutNFinN{\displaystyle \mathrm {MA} _{\text{compound}}={F_{\text{out1}} \over F_{\text{in1}}}{F_{\text{out2}} \over F_{\text{in2}}}{F_{\text{out3}} \over F_{\text{in3}}}\ldots {F_{{\text{out}}N} \over F_{{\text{in}}N}}\,}

Thus, the mechanical advantage of the compound machine is equal to the product of the mechanical advantages of the series of simple machines that form it

MAcompound=MA1MA2…MAN{\displaystyle \mathrm {MA} _{\text{compound}}=\mathrm {MA} _{1}\mathrm {MA} _{2}\ldots \mathrm {MA} _{N}}

Similarly, the efficiency of a compound machine is also the product of the efficiencies of the series of simple machines that form it

ηcompound=η1η2…ηN.{\displaystyle \eta _{\text{compound}}=\eta _{1}\eta _{2}\ldots \;\eta _{N}.}

Self-locking machines

Thescrew's self-locking property is the reason for its wide use inthreaded fasteners likebolts andwood screws

In many simple machines, if the load forceFout{\displaystyle F_{\textrm {out}}} on the machine is high enough in relation to the input forceFin{\displaystyle F_{\textrm {in}}}, the machine will move backwards, with the load force doing work on the input force.[31] So these machines can be used in either direction, with the driving force applied to either input point. For example, if the load force on a lever is high enough, the lever will move backwards, moving the input arm backwards against the input force. These are calledreversible,non-locking oroverhauling machines, and the backward motion is calledoverhauling.

However, in some machines, if the frictional forces are high enough, no amount of load force can move it backwards, even if the input force is zero. This is called aself-locking,nonreversible, ornon-overhauling machine.[31] These machines can only be set in motion by a force at the input, and when the input force is removed will remain motionless, "locked" by friction at whatever position they were left.

Self-locking occurs mainly in those machines with large areas of sliding contact between moving parts: thescrew,inclined plane, andwedge:

  • The most common example is a screw. In most screws, one can move the screw forward or backward by turning it, and one can move the nut along the shaft by turning it, but no amount of pushing the screw or the nut will cause either of them to turn.
  • On an inclined plane, a load can be pulled up the plane by a sideways input force, but if the plane is not too steep and there is enough friction between load and plane, when the input force is removed the load will remain motionless and will not slide down the plane, regardless of its weight.
  • A wedge can be driven into a block of wood by force on the end, such as from hitting it with a sledge hammer, forcing the sides apart, but no amount ofcompression force from the wood walls will cause it to pop back out of the block.

A machine will be self-locking if and only if its efficiencyη{\displaystyle \eta } is below 50%:[31]

η≡Fout/Findin/dout<0.5{\displaystyle \eta \equiv {\frac {F_{\text{out}}/F_{\text{in}}}{d_{\text{in}}/d_{\text{out}}}}<0.5}

Whether a machine is self-locking depends on both the friction forces (coefficient of static friction) between its parts, and the distance ratiodin/dout{\displaystyle d_{\textrm {in}}/d_{\textrm {out}}} (ideal mechanical advantage). If both the friction and ideal mechanical advantage are high enough, it will self-lock.

Proof

When a machine moves in the forward direction from point 1 to point 2, with the input force doing work on a load force, from conservation of energy[32][33] the input workW1,2{\displaystyle W_{\text{1,2}}} is equal to the sum of the work done on the load forceWload{\displaystyle W_{\text{load}}} and the work lost to frictionWfric{\displaystyle W_{\text{fric}}}

W1,2=Wload+Wfric{\displaystyle W_{\text{1,2}}=W_{\text{load}}+W_{\text{fric}}}Eq. 1

If the efficiency is below 50%(η=Wload/W1,2<0.5{\displaystyle \eta =W_{\text{load}}/W_{\text{1,2}}<0.5}):

2Wload<W1,2{\displaystyle 2W_{\text{load}}<W_{\text{1,2}}\,}

FromEq. 12Wload<Wload+WfricWload<Wfric{\displaystyle {\begin{aligned}2W_{\text{load}}&<W_{\text{load}}+W_{\text{fric}}\\W_{\text{load}}&<W_{\text{fric}}\end{aligned}}}

When the machine moves backward from point 2 to point 1 with the load force doing work on the input force, the work lost to frictionWfric{\displaystyle W_{\text{fric}}} is the same

Wload=W2,1+Wfric{\displaystyle W_{\text{load}}=W_{\text{2,1}}+W_{\text{fric}}}

So the output work isW2,1=Wload−Wfric<0{\displaystyle W_{\text{2,1}}=W_{\text{load}}-W_{\text{fric}}<0}

Thus the machine self-locks, because the work dissipated in friction is greater than the work done by the load force moving it backwards even with no input force.

Modern machine theory

Machines are studied as mechanical systems consisting ofactuators andmechanisms that transmit forces and movement, monitored by sensors and controllers. The components of actuators and mechanisms consist of links and joints that form kinematic chains.

Kinematic chains

Illustration of a Four-bar linkage from Kinematics of Machinery, 1876
Illustration of a four-bar linkage fromKinematics of Machinery, 1876

Simple machines are elementary examples ofkinematic chains that are used to modelmechanical systems ranging from the steam engine to robot manipulators. The bearings that form the fulcrum of a lever and that allow the wheel and axle and pulleys to rotate are examples of akinematic pair called a hinged joint. Similarly, the flat surface of an inclined plane and wedge are examples of the kinematic pair called a sliding joint. The screw is usually identified as its own kinematic pair called a helical joint.

Two levers, or cranks, are combined into a planarfour-bar linkage by attaching a link that connects the output of one crank to the input of another. Additional links can be attached to form asix-bar linkage or in series to form a robot.[26]

Classification of machines

The identification of simple machines arises from a desire for a systematic method to invent new machines. Therefore, an important concern is how simple machines are combined to make more complex machines. One approach is to attach simple machines in series to obtain compound machines.

However, a more successful strategy was identified byFranz Reuleaux, who collected and studied over 800 elementary machines. He realized that a lever, pulley, and wheel and axle are in essence the same device: a body rotating about a hinge. Similarly, an inclined plane, wedge, and screw are a block sliding on a flat surface.[34]

This realization shows that it is the joints, or the connections that provide movement, that are the primary elements of a machine. Starting with four types of joints, therevolute joint,sliding joint,cam joint andgear joint, and related connections such as cables and belts, it is possible to understand a machine as an assembly of solid parts that connect these joints.[26]

Kinematic synthesis

The design of mechanisms to perform required movement and force transmission is known askinematic synthesis. This is a collection of geometric techniques for the mechanical design oflinkages,cam and follower mechanisms andgears and gear trains.

See also

References

  1. ^Paul, Akshoy; Roy, Pijush; Mukherjee, Sanchayan (2005),Mechanical sciences: engineering mechanics and strength of materials, Prentice Hall of India, p. 215,ISBN 978-81-203-2611-8.
  2. ^abcAsimov, Isaac (1988),Understanding Physics, New York: Barnes & Noble, p. 88,ISBN 978-0-88029-251-1.
  3. ^Anderson, William Ballantyne (1914).Physics for Technical Students: Mechanics and Heat. New York: McGraw Hill. p. 112. RetrievedMay 11, 2008.
  4. ^"Mechanics".Encyclopædia Britannica. Vol. 3. John Donaldson. 1773. p. 44. RetrievedApril 5, 2020.
  5. ^Morris, Christopher G. (1992).Academic Press Dictionary of Science and Technology. Gulf Professional Publishing. p. 1993.ISBN 978-0122004001.
  6. ^abCompound machines, University of Virginia Physics Department, retrievedJune 11, 2010.
  7. ^abUsher, Abbott Payson (1988).A History of Mechanical Inventions. US: Courier Dover Publications. p. 98.ISBN 978-0-486-25593-4.
  8. ^Wallenstein, Andrew (June 2002)."Foundations of cognitive support: Toward abstract patterns of usefulness".Proceedings of the 9th Annual Workshop on the Design, Specification, and Verification of Interactive Systems. Springer. p. 136.ISBN 978-3540002666. RetrievedMay 21, 2008.
  9. ^abPrater, Edward L. (1994),Basic machines(PDF), U.S. Navy Naval Education and Training Professional Development and Technology Center, NAVEDTRA 14037.
  10. ^U.S. Navy Bureau of Naval Personnel (1971),Basic machines and how they work(PDF), Dover Publications.
  11. ^Reuleaux, F. (1963) [1876],The kinematics of machinery (translated and annotated by A.B.W. Kennedy), New York: reprinted by Dover.
  12. ^Cornell University,Reuleaux Kinematic Mechanisms Collection, Cornell University.
  13. ^abChiu, Y. C. (2010),An introduction to the History of Project Management, Delft: Eburon Academic Publishers, p. 42,ISBN 978-90-5972-437-2
  14. ^Ostdiek, Vern; Bord, Donald (2005).Inquiry into Physics. Thompson Brooks/Cole. p. 123.ISBN 978-0-534-49168-0. RetrievedMay 22, 2008.
  15. ^Quoted byPappus of Alexandria inSynagoge, Book VIII
  16. ^Dupac, Mihai; Marghitu, Dan B. (2021).Engineering Applications: Analytical and Numerical Calculation with MATLAB. John Wiley and Sons. p. 295.ISBN 9781119093633.
  17. ^Dijksterhuis, Eduard Jan (2014).Archimedes. Princeton University Press. p. 15.ISBN 9781400858613.
  18. ^Strizhak, Viktor; Igor Penkov; Toivo Pappel (2004)."Evolution of design, use, and strength calculations of screw threads and threaded joints".HMM2004 International Symposium on History of Machines and Mechanisms. Kluwer Academic. p. 245.ISBN 1-4020-2203-4. RetrievedMay 21, 2008.
  19. ^abKrebs, Robert E. (2004).Groundbreaking Experiments, Inventions, and Discoveries of the Middle Ages. Greenwood. p. 163.ISBN 978-0-313-32433-8. RetrievedMay 21, 2008.
  20. ^Stephen, Donald; Lowell Cardwell (2001).Wheels, clocks, and rockets: a history of technology. US: W. W. Norton & Company. pp. 85–87.ISBN 978-0-393-32175-3.
  21. ^Armstrong-Hélouvry, Brian (1991).Control of machines with friction. Springer. p. 10.ISBN 978-0-7923-9133-3.
  22. ^This fundamental insight was the subject of Galileo Galilei's 1600 workLe Meccaniche (On Mechanics).
  23. ^abBhatnagar, V. P. (1996).A Complete Course in Certificate Physics. India: Pitambar. pp. 28–30.ISBN 978-81-209-0868-0.
  24. ^Simmons, Ron; Cindy, Barden (2008).Discover! Work & Machines. US: Milliken. p. 29.ISBN 978-1-4291-0947-5.
  25. ^Gujral, I. S. (2005).Engineering Mechanics. Firewall Media. pp. 378–380.ISBN 978-81-7008-636-9.
  26. ^abcUicker, John J. Jr.; Pennock, Gordon R.; Shigley, Joseph E. (2003),Theory of Machines and Mechanisms (third ed.), New York: Oxford University Press,ISBN 978-0-19-515598-3
  27. ^Paul, Burton (1979).Kinematics and Dynamics of Planar Machinery. Prentice Hall.ISBN 978-0-13-516062-6.
  28. ^Rao, S.; Durgaiah, R. (2005).Engineering Mechanics. Universities Press. p. 80.ISBN 978-81-7371-543-3.
  29. ^Goyal, M. C.; Raghuvanshee, G. S. (2011).Engineering Mechanics. PHI Learning. p. 212.ISBN 978-81-203-4327-6.
  30. ^Avison, John (2014).The World of Physics. Nelson Thornes. p. 110.ISBN 978-0-17-438733-6.
  31. ^abcGujral, I. S. (2005).Engineering Mechanics. Firewall Media. p. 382.ISBN 978-81-7008-636-9.
  32. ^Rao, S.; Durgaiah, R. (2005).Engineering Mechanics. Universities Press. p. 82.ISBN 978-81-7371-543-3.
  33. ^Goyal, M. C.; Raghuvanshi, G. S. (2009).Engineering Mechanics. New Delhi: PHI Learning Private Ltd. p. 202.ISBN 978-81-203-3789-3.
  34. ^Hartenberg, R.S. & J. Denavit (1964)Kinematic synthesis of linkages, New York: McGraw-Hill, online link fromCornell University.
Classical simple machines
Clocks
Compressors andpumps
External combustion engines
Internal combustion engines
Linkages
Turbine
Aerofoil
Electronics
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